15 IoT Mathematics: Linear Algebra and Probability

15.8.1 C.1 Vectors

Notation: $\vec{v} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ or simply $[x, y, z]$

IoT Applications:

Vector Operations:

Magnitude (Length): How “big” is the vector? $|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$

Example: Accelerometer reads $[0.5, 0.3, 0.8]g$ $|\vec{a}| = \sqrt{0.5^2 + 0.3^2 + 0.8^2} = \sqrt{0.98} \approx 0.99g$

This tells us the total acceleration magnitude (useful for fall detection).

15.8.2 C.2 Matrices

Notation: $M = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$

IoT Applications:

1. Coordinate Transformation:

Rotating sensor data to align with a reference frame:

2. Kalman Filter State:

State vector might be $[position, velocity, acceleration]$: $\vec{x} = \begin{bmatrix} p \\ v \\ a \end{bmatrix}$

State transition predicts next state: $\vec{x}_{k+1} = F \cdot \vec{x}_k$

3. Covariance Matrix:

Describes uncertainty in measurements: $P = \begin{bmatrix} \sigma_x^2 & \sigma_{xy} \\ \sigma_{xy} & \sigma_y^2 \end{bmatrix}$

Diagonal elements: Variance (uncertainty) in each variable Off-diagonal: How variables correlate (change together)

15.8.3 C.3 Matrix Multiplication

Example: Transforming a 3D point $\begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} = \begin{bmatrix} 2 \\ 4 \\ 6 \end{bmatrix}$

This scales the vector by 2 in all directions.

15.9.1 D.1 Basic Probability

Key Concepts:

Practical Example - Sensor Reliability:

If a sensor has 99.9% uptime:

• P(working) = 0.999
• P(failure) = 0.001

For 100 sensors with 99.9% uptime each, the expected number unavailable at a randomly chosen instant is 100 x 0.001 = 0.1.

15.9.2 D.2 Gaussian (Normal) Distribution

$p(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

Properties:

• 68% of values within mu +/- 1 sigma
• 95% of values within mu +/- 2 sigma
• 99.7% of values within mu +/- 3 sigma

IoT Application - Sensor Noise:

A temperature sensor with:

• Mean reading: mu = 25.0 C
• Noise: sigma = 0.5 C

95% of readings will be between 24.0 C and 26.0 C.

15.9.3 D.3 Sensor Fusion: Combining Measurements

When you have two noisy measurements of the same thing, the optimal combination is:

$x_{combined} = \frac{\sigma_2^2 \cdot x_1 + \sigma_1^2 \cdot x_2}{\sigma_1^2 + \sigma_2^2}$

Intuition: Weight each measurement inversely by its uncertainty. Trust the more precise sensor more.

Example:

• GPS says position is 100m (sigma = 10m)
• Accelerometer integration says 95m (sigma = 5m)

$x_{combined} = \frac{25 \times 100 + 100 \times 95}{100 + 25} = \frac{2500 + 9500}{125} = 96m$

The combined estimate is closer to the accelerometer (more precise).

15.10.1 E.1 The Modulo Operation

Modulo is the arithmetic mechanism behind the wraparound behavior used throughout this section. Begin with small remainders to verify the operation, then carry the same rule into modular exponentiation. The examples establish the computation; later cryptographic sections add the essential requirements for large parameters, randomness, authentication, and safe key handling.

Notation: $a \mod n$ = remainder when a is divided by n

Examples:

• $7 \mod 5 = 2$ (7 = 1x5 + 2)
• $25 \mod 7 = 4$ (25 = 3x7 + 4)
• $12 \mod 12 = 0$ (clock wraps at midnight)

15.10.2 E.2 Why Modular Arithmetic for Cryptography?

Key Property of modular exponentiation in suitable large groups: Easy to compute forward; the discrete logarithm can be hard to recover.

Example - Discrete Logarithm:

Given: $g = 5$, $p = 23$

Forward (easy): Calculate $5^7 \mod 23$ $5^7 = 78125 \mod 23 = 17$

Reverse (hard only with suitable large parameters; easy in this toy example): Given 17, find x where $5^x \mod 23 = 17$

This one-way property supports Diffie-Hellman key agreement and related public-key systems.

15.10.3 E.3 Diffie-Hellman Key Exchange

Two IoT devices can agree on a shared secret over an insecure channel. First read the message sequence in Figure 15.1 from public parameters through exchanged public values to the independently derived shared result.

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Inspect Figure 15.2 to ask how two devices can exchange public mixtures yet reach the same result without sending either private part.

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Read Figure 15.2 from the public yellow to the final brown. One device mixes in private red and sends orange. The other mixes in private blue and sends green. Each device adds its own secret colour to the mix it got, so both reach brown. A watcher sees yellow, orange, and green but cannot split either mix to find red or blue. In the math, values such as 8 and 19 may be public, while the private powers stay hard to find when the numbers are large. The picture explains a one-way step, not peer trust; real links still need known peers and safe key care.

15.11.1 F.1 What Are Complex Numbers?

Form: $z = a + bi$

• $a$ = real part
• $b$ = imaginary part

Alternative (Polar) Form: $z = r \cdot e^{i\theta}$

• $r$ = magnitude (amplitude)
• $\theta$ = phase angle

15.11.2 F.2 Why Complex Numbers in IoT?

1. Representing Sinusoidal Signals:

A sensor reading: $v(t) = A \cos(\omega t + \phi)$

Using Euler’s formula: $e^{i\theta} = \cos\theta + i\sin\theta$

We can write: $v(t) = \text{Re}[A \cdot e^{i(\omega t + \phi)}]$

2. Fourier Transform:

Converts time-domain signals to frequency-domain: $X(f) = \int_{-\infty}^{\infty} x(t) \cdot e^{-i2\pi ft} \, dt$

The result is complex:

• Magnitude $|X(f)|$ tells you “how much” of each frequency
• Phase $\angle X(f)$ tells you “when” each frequency component peaks

3. I/Q Signals in Radio:

LoRa and other radios use I/Q (In-phase/Quadrature) representation:

• I = real part of signal
• Q = imaginary part of signal

This allows representing any modulation scheme.

15.11.3 F.3 Complex Number Operations

15.12.1 G.1 Bits and Entropy

Treat bits and entropy as measures of uncertainty, not synonyms for file size. First identify the possible outcomes and their probabilities, then calculate how much information a result conveys. The distinction connects source data, compression, cryptographic randomness, and channel capacity while keeping each claim tied to its own assumptions.

Entropy formula: $H = -\sum_i p_i \log_2(p_i)$

Where $p_i$ is the probability of each outcome.

Example - Sensor Alarm:

• Alarm triggers 1% of the time
• $H = -(0.01 \log_2 0.01 + 0.99 \log_2 0.99)$
• $H = -(0.01 \times -6.64 + 0.99 \times -0.014) = 0.08$ bits

The alarm provides only 0.08 bits of information per reading (very predictable).

15.12.2 G.2 Shannon Capacity

The maximum rate at which information can be reliably transmitted:

$C = B \log_2(1 + SNR)$

Practical Implications:

Key Insight: To double capacity, you need to roughly quadruple SNR. This explains why LoRa trades data rate for range (lower SNR = lower rate but still works).

15.13.1 Essential Formulas for IoT